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Integer and Combinatorial Optimization.

By: Contributor(s): Series: Wiley Series in Discrete Mathematics and Optimization SerPublisher: Somerset : John Wiley & Sons, Incorporated, 1988Copyright date: ©1988Edition: 1st edDescription: 1 online resource (783 pages)Content type:
  • text
Media type:
  • computer
Carrier type:
  • online resource
ISBN:
  • 9781118626863
Subject(s): Genre/Form: Additional physical formats: Print version:: Integer and Combinatorial OptimizationDDC classification:
  • 519.7/7
LOC classification:
  • QA402.5 -- .N464 1999eb
Online resources:
Contents:
Cover -- Title Page -- Copyright -- Contents -- Preface -- Part I: Foundations -- I.1 The Scope of Integer and Combinatorial Optimization -- 1. Introduction -- 2. Modeling with Binary Variables I: Knapsack, Assignmentand Matching, Covering, Packing and Partitioning -- The 0-1 Knapsack Problem -- The Assignment and Matching Problems -- Set-covering, Set-packing, and Set-partitioning Problems -- 3. Modeling with Binary Variables II: Facility Location, Fixed-charge Network Flow, and Traveling Salesman -- Facility Location Problems -- The Fixed-charge Network Flow Problem -- The Traveling Salesman Problem -- 4. Modeling with Binary Variables III: Nonlinear Functions and Disjunctive Constraints -- Piecewise Linear Functions -- Disjunctive Constraints -- A Scheduling Problem -- 5. Choices in Model Formulation -- 6. Preprocessing -- Tightening Bounds -- Adding Logical Inequalities, Fixing Variables, and Removing Redundant Constraints -- 7. Notes -- Section I.1.1 -- Sections I.1.2-I.1.4 -- Section I.1.5 -- Section I.1.6 -- 8. Exercises -- I.2: Linear Programming -- 1. Introduction -- 2. Duality -- 3. The Primal and Dual Simplex Algorithms -- Bases and Basic Solutions -- Changing the Basis -- Primal Simplex Algorithm -- Dual Simplex Algorithm -- Dual Simplex Algorithm (phase 2) -- The Simplex Algorithm with Simple Upper Bounds -- Addition of Constraints or Variables -- 4. Subgradient Optimization -- The Subgradient Algorithm for (4.1) -- 5. Notes -- Sections I.2.1-i.2.3. -- Section I.2.4 -- I.3: Graphs and Networks -- 1. Introduction -- 2. The Minimum-weight or Shortest-path Problem -- Dijkstra's Minimum-weight Path Algorithm -- Bellman-ford Minimum-weight Path Algorithm -- 3. The Minimum-weight Spanning Tree Problem -- Algorithm for Constructing a Spanning Tree -- 4. The Maximum-flow and Minimum-cut Problems -- Augmenting Path Algorithm.
5. The Transportation Problem: A Primal-dual Algorithm -- Primal-dual Algorithm for the Transportation Problem -- Minimum-cost Path Augmentation Algorithm -- 6. A Primal Simplex Algorithm for Network Flow Problems -- 7. Notes -- Section I.3.1 -- Section I.3.2 -- Section I.3.3 -- Section I.3.4 -- Section I.3.5 -- Section I.3.6 -- I.4: Polyhedral Theory -- 1. Introduction and Elementary Linear Algebra -- 2. Definitions of Polyhedra and Dimension -- 3. Describing Polyhedra by Facets -- 4. Describing Polyhedra by Extreme Points and Extreme Rays -- 5. Polarity -- 6. Polyhedral Ties Between Linear and Integer Programs -- 7. Notes -- Sections I.4.1-I.4.4 -- Section I.4.5 -- Section I.4.6 -- 8. Exercises -- 1.5: Computational Complexity -- 1. Introduction -- 2. Measuring Algorithm Efficiency and Problem Complexity -- 3. Some Problems Solvable in Polynomial Time -- 4. Remarks on 0-1 and Pure-integer Programming -- 5. Nondeterministic Polynomial-time Algorithms and Np Problems -- Certificates of Feasibility, the Class Np, and Nondeterministic Algorithms -- 6. The Most Difficult Np Problems: the Class Np -- 7. Complexity and Polyhedra -- 8. Notes -- Sections I.5.1 and I.5.2 -- Section I.5.3 -- Section I.5.4 -- Sections I.5.5 and I.5.6 -- Section I.5.7 -- 9. Exercises -- I.6: Polynomial-Time Algorithms for Linear Programming -- 1. Introduction -- 2. The Ellipsoid Algorithm -- The Ellipsoid Algorithm for P< -- 3. The Polynomial Equivalence of Separation and Optimization -- 4. A Projective Algorithm for Linear Programming -- The Projective Algorithm to Find an E-approximate Ray -- The Projective Algorithm for the Linear Program (4.2) -- 5. A Strongly Polynomial Algorithm for Combinatorial Linear Programs -- 6. Notes -- Section I.6.1 -- Section I.6.2 -- Section I.6.3 -- Section I.6.4 -- Section I.6.5 -- I.7: Integer Lattices -- 1. Introduction.
2. The Euclidean Algorithm -- The Euclidean Algorithm to Find Gcd(a, b) -- 3. Continued Fractions -- 4. Lattices and Hermite Normal Form -- The Hermite Normal Form Algorithm -- 5. A Reduced Basis of a Lattice -- The Gram-schmidt Orthogonalization of a Basis B -- A Reduced Basis Algorithm for a Full-dimensional Lattice L -- 6. Notes -- Section I.7.1 -- Section I.7.2 -- Section I.7.3 -- Section I.7.4 -- Section I.7.5 -- 7. Exercises -- Part II: General Integer Programming -- II. 1: the Theory of Valid Inequalities -- 1. Introduction -- 2. Generating All Valid Inequalities -- 0-1 Problems -- Bounded Integer Variables -- 3. Gomory's Fractional Cuts and Rounding -- 4. Superadditive Functions and Valid Inequalities -- 5. A Polyhedral Description of Superadditive Valid Inequalities for Independence Systems -- 6. Valid Inequalities for Mixed-integer Sets -- Mixed-integer Rounding (mir) Procedure -- 7. Superadditivity for Mixed-integer Sets -- 8. Notes -- Section II.1.1 -- Section II.1.2 -- Section II.1.3 -- Section II.1.4 -- Section II.1.5 -- Section II.1.6 -- Section II.1.7 -- 9. Exercises -- II.2: Strong Valid Inequalities and Facets for Structured Integer Programs -- 1. Introduction -- 2. Valid Inequalities for the 0-1 Knapsack Polytope -- 3. Valid Inequalities for the Symmetric Traveling Salesman Polytope -- 4. Valid Inequalities for Variable Upper-bound Flow Models -- 5. Notes -- Section II.2.1 -- Section II.2.2 -- Section II.2.3 -- Section II.2.4 -- 6. Exercises -- II.3: Duality and Relaxation -- 1. Introduction -- 2. Duality and the Value Function -- 3. Superadditive Duality -- 4. The Maximum-weight Path Formulation and Superadditive Duality -- 5. Modular Arithmetic and the Group Problem -- 6. Lagrangian Relaxation and Duality -- 7. Benders' Reformulation -- 8. Notes -- Section II.3.1 -- Section II.3.2 -- Section II.3.3 -- Section II.3.4.
Section II.3.5 -- Section II.3.6 -- Section II.3.7 -- 9. Exercises -- II.4: General Algorithms -- 1. Introduction -- General Relaxation Algorithm -- Fractional Cutting-plane Algorithm (fcpa) -- General Branch-and-bound Algorithm -- 2. Branch-and-bound Using Linear Programming Relaxations -- Pruning Criteria -- Division -- Node Selection -- Branching Variable Selection -- Generalized Upper-bound Constraints -- Piecewise Linear Functions -- 3. General Cutting-plane Algorithms -- Finite Convergence -- The Lexicographic Dual Simplex Algorithm -- Extension to Mixed-integer Programming -- Primal Cutting-plane Algorithm -- 4. Notes -- Section II.4.1 -- Section II.4.2 -- Section II.4.3 -- 5. Exercises -- II.5: Special-purpose Algorithms -- 1. Introduction -- 2. A Cuiting-plane Algorithm Using Strong Valid Inequalities -- Fractional Cutting-plane Algorithm (fcpa) for Lp(f) -- The Separation Problem for F -- A Strong Cutting-plane/branch-and-bound Algorithm for Ip. -- 3. Primal and Dual Heuristic Algorithms -- A Greedy (heuristic) Algorithm for Maximizing a Set Function -- A K-interchange Heuristic for Max{c(x): X Є S e Bn}. -- Dual Descent [a Greedy Algorithm for (3.2)] -- Analysis of Heuristics -- Simulated Annealing -- Simulated Annealing Algorithm for (3.5) -- Probabilistic Analysis -- 4. Decomposition Algorithms -- Solving the Lagrangian Dual by Subgradient Optimization -- Solving the Lagrangian Dual by Constraint Generation -- Benders' Decomposition -- Constraint Generation Algorithm for Mip' -- 5. Dynamic Programming -- A Dynamic Programming Algorithm for the 0-1 Knapsack Problem -- A Dynamic Programming Algorithm for the Uncapacitated Lot-size Problem (uls) -- 6. Notes -- Section II.5.1 -- Section II.5.2 -- Section II.5.3 -- Section II.5.4 -- Section II.5.5 -- 7. Exercises -- II.6: Applications of Special-purpose Algorithms.
1. Knapsack and Group Problems -- The Integer Knapsack Problem -- Dynamic Programming -- A Superadditive Dual Algorithm -- Heuristic Algorithms -- The Scaling/rounding (sr) Heuristic -- The Group Problem -- A Shortest-path Enumeration Algorithm for Ip -- The Increasing Group Algorithm -- The 0-1 Knapsack Problem -- An Algorithm for the Linear Programming Relaxation -- Primal Heuristic Algorithm -- Branch-and-bound -- 2. 0-1 Integer Programming Problems -- A Simplex-based Heuristic for Bip -- Algorithm -- An Fcp/branch-and-bound Algorithm -- Set Covering and Packing -- Greedy Heuristic for Set Covering -- 3. The Symmetric Traveling Salesman Problem -- Relaxations -- Primal Heuristics -- Relaxation/branch-and-bound Algorithms -- Shrinking an Edge E = (i,j) of G(x*) with Xe*= 1 -- Solution of a Large Problem -- 4. Fixed-charge Network Flow Problems -- A Branch-and-bound Algorithm for Fn -- An Fcpa for Fn -- Separation Algorithm for Generalized Flow Cover Inequalities -- Solving a Fixed-charge Uncapacitated Transportation Problem by an Fcpa and Branch-and-bound -- A Reformulation of the Single Source Problem (sfn) -- Steiner Branchings -- Reformulations Ofthe Uncapacitated Lot-size Problem (uls) -- 5. Applications of Basis Reduction -- The Subset Sum Problem -- The Reduced Basis Algorithm to Find a Solution of (5.1) -- The Linear Inequality Integer Feasibility Problem -- 6. Notes -- Section II.6.1 -- Section II.6.2 -- Section II.6.3 -- Section II.6.4 -- Section II.6.5 -- 7. Exercises -- Part III: Combinatorial Optimization -- III.1: Integral Polyhedra -- 1. Introduction -- 2. Totally Unimodular Matrices -- 3. Network Matrices -- Algorithm for Recognizing an Ept Matrix -- 4. Balanced and Totally Balanced Matrices -- Algorithm for Dfc and Fc for Row Inclusion Matrices -- Algorithm for Recognizing Tb Matrices -- 5. Node Packing and Perfect Graphs.
Algorithm for Dfnp and Fnp for Chordal Graphs.
Summary: Rave reviews for INTEGER AND COMBINATORIAL OPTIMIZATION "This book provides an excellent introduction and survey of traditional fields of combinatorial optimization . . . It is indeed one of the best and most complete texts on combinatorial optimization . . . available. [And] with more than 700 entries, [it] has quite an exhaustive reference list."-Optima "A unifying approach to optimization problems is to formulate them like linear programming problems, while restricting some or all of the variables to the integers. This book is an encyclopedic resource for such formulations, as well as for understanding the structure of and solving the resulting integer programming problems."-Computing Reviews "[This book] can serve as a basis for various graduate courses on discrete optimization as well as a reference book for researchers and practitioners."-Mathematical Reviews "This comprehensive and wide-ranging book will undoubtedly become a standard reference book for all those in the field of combinatorial optimization."-Bulletin of the London Mathematical Society "This text should be required reading for anybody who intends to do research in this area or even just to keep abreast of developments."-Times Higher Education Supplement, London Also of interest . . . INTEGER PROGRAMMING Laurence A. Wolsey Comprehensive and self-contained, this intermediate-level guide to integer programming provides readers with clear, up-to-date explanations on why some problems are difficult to solve, how techniques can be reformulated to give better results, and how mixed integer programming systems can be used more effectively. 1998 (0-471-28366-5) 260 pp.
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Cover -- Title Page -- Copyright -- Contents -- Preface -- Part I: Foundations -- I.1 The Scope of Integer and Combinatorial Optimization -- 1. Introduction -- 2. Modeling with Binary Variables I: Knapsack, Assignmentand Matching, Covering, Packing and Partitioning -- The 0-1 Knapsack Problem -- The Assignment and Matching Problems -- Set-covering, Set-packing, and Set-partitioning Problems -- 3. Modeling with Binary Variables II: Facility Location, Fixed-charge Network Flow, and Traveling Salesman -- Facility Location Problems -- The Fixed-charge Network Flow Problem -- The Traveling Salesman Problem -- 4. Modeling with Binary Variables III: Nonlinear Functions and Disjunctive Constraints -- Piecewise Linear Functions -- Disjunctive Constraints -- A Scheduling Problem -- 5. Choices in Model Formulation -- 6. Preprocessing -- Tightening Bounds -- Adding Logical Inequalities, Fixing Variables, and Removing Redundant Constraints -- 7. Notes -- Section I.1.1 -- Sections I.1.2-I.1.4 -- Section I.1.5 -- Section I.1.6 -- 8. Exercises -- I.2: Linear Programming -- 1. Introduction -- 2. Duality -- 3. The Primal and Dual Simplex Algorithms -- Bases and Basic Solutions -- Changing the Basis -- Primal Simplex Algorithm -- Dual Simplex Algorithm -- Dual Simplex Algorithm (phase 2) -- The Simplex Algorithm with Simple Upper Bounds -- Addition of Constraints or Variables -- 4. Subgradient Optimization -- The Subgradient Algorithm for (4.1) -- 5. Notes -- Sections I.2.1-i.2.3. -- Section I.2.4 -- I.3: Graphs and Networks -- 1. Introduction -- 2. The Minimum-weight or Shortest-path Problem -- Dijkstra's Minimum-weight Path Algorithm -- Bellman-ford Minimum-weight Path Algorithm -- 3. The Minimum-weight Spanning Tree Problem -- Algorithm for Constructing a Spanning Tree -- 4. The Maximum-flow and Minimum-cut Problems -- Augmenting Path Algorithm.

5. The Transportation Problem: A Primal-dual Algorithm -- Primal-dual Algorithm for the Transportation Problem -- Minimum-cost Path Augmentation Algorithm -- 6. A Primal Simplex Algorithm for Network Flow Problems -- 7. Notes -- Section I.3.1 -- Section I.3.2 -- Section I.3.3 -- Section I.3.4 -- Section I.3.5 -- Section I.3.6 -- I.4: Polyhedral Theory -- 1. Introduction and Elementary Linear Algebra -- 2. Definitions of Polyhedra and Dimension -- 3. Describing Polyhedra by Facets -- 4. Describing Polyhedra by Extreme Points and Extreme Rays -- 5. Polarity -- 6. Polyhedral Ties Between Linear and Integer Programs -- 7. Notes -- Sections I.4.1-I.4.4 -- Section I.4.5 -- Section I.4.6 -- 8. Exercises -- 1.5: Computational Complexity -- 1. Introduction -- 2. Measuring Algorithm Efficiency and Problem Complexity -- 3. Some Problems Solvable in Polynomial Time -- 4. Remarks on 0-1 and Pure-integer Programming -- 5. Nondeterministic Polynomial-time Algorithms and Np Problems -- Certificates of Feasibility, the Class Np, and Nondeterministic Algorithms -- 6. The Most Difficult Np Problems: the Class Np -- 7. Complexity and Polyhedra -- 8. Notes -- Sections I.5.1 and I.5.2 -- Section I.5.3 -- Section I.5.4 -- Sections I.5.5 and I.5.6 -- Section I.5.7 -- 9. Exercises -- I.6: Polynomial-Time Algorithms for Linear Programming -- 1. Introduction -- 2. The Ellipsoid Algorithm -- The Ellipsoid Algorithm for P< -- 3. The Polynomial Equivalence of Separation and Optimization -- 4. A Projective Algorithm for Linear Programming -- The Projective Algorithm to Find an E-approximate Ray -- The Projective Algorithm for the Linear Program (4.2) -- 5. A Strongly Polynomial Algorithm for Combinatorial Linear Programs -- 6. Notes -- Section I.6.1 -- Section I.6.2 -- Section I.6.3 -- Section I.6.4 -- Section I.6.5 -- I.7: Integer Lattices -- 1. Introduction.

2. The Euclidean Algorithm -- The Euclidean Algorithm to Find Gcd(a, b) -- 3. Continued Fractions -- 4. Lattices and Hermite Normal Form -- The Hermite Normal Form Algorithm -- 5. A Reduced Basis of a Lattice -- The Gram-schmidt Orthogonalization of a Basis B -- A Reduced Basis Algorithm for a Full-dimensional Lattice L -- 6. Notes -- Section I.7.1 -- Section I.7.2 -- Section I.7.3 -- Section I.7.4 -- Section I.7.5 -- 7. Exercises -- Part II: General Integer Programming -- II. 1: the Theory of Valid Inequalities -- 1. Introduction -- 2. Generating All Valid Inequalities -- 0-1 Problems -- Bounded Integer Variables -- 3. Gomory's Fractional Cuts and Rounding -- 4. Superadditive Functions and Valid Inequalities -- 5. A Polyhedral Description of Superadditive Valid Inequalities for Independence Systems -- 6. Valid Inequalities for Mixed-integer Sets -- Mixed-integer Rounding (mir) Procedure -- 7. Superadditivity for Mixed-integer Sets -- 8. Notes -- Section II.1.1 -- Section II.1.2 -- Section II.1.3 -- Section II.1.4 -- Section II.1.5 -- Section II.1.6 -- Section II.1.7 -- 9. Exercises -- II.2: Strong Valid Inequalities and Facets for Structured Integer Programs -- 1. Introduction -- 2. Valid Inequalities for the 0-1 Knapsack Polytope -- 3. Valid Inequalities for the Symmetric Traveling Salesman Polytope -- 4. Valid Inequalities for Variable Upper-bound Flow Models -- 5. Notes -- Section II.2.1 -- Section II.2.2 -- Section II.2.3 -- Section II.2.4 -- 6. Exercises -- II.3: Duality and Relaxation -- 1. Introduction -- 2. Duality and the Value Function -- 3. Superadditive Duality -- 4. The Maximum-weight Path Formulation and Superadditive Duality -- 5. Modular Arithmetic and the Group Problem -- 6. Lagrangian Relaxation and Duality -- 7. Benders' Reformulation -- 8. Notes -- Section II.3.1 -- Section II.3.2 -- Section II.3.3 -- Section II.3.4.

Section II.3.5 -- Section II.3.6 -- Section II.3.7 -- 9. Exercises -- II.4: General Algorithms -- 1. Introduction -- General Relaxation Algorithm -- Fractional Cutting-plane Algorithm (fcpa) -- General Branch-and-bound Algorithm -- 2. Branch-and-bound Using Linear Programming Relaxations -- Pruning Criteria -- Division -- Node Selection -- Branching Variable Selection -- Generalized Upper-bound Constraints -- Piecewise Linear Functions -- 3. General Cutting-plane Algorithms -- Finite Convergence -- The Lexicographic Dual Simplex Algorithm -- Extension to Mixed-integer Programming -- Primal Cutting-plane Algorithm -- 4. Notes -- Section II.4.1 -- Section II.4.2 -- Section II.4.3 -- 5. Exercises -- II.5: Special-purpose Algorithms -- 1. Introduction -- 2. A Cuiting-plane Algorithm Using Strong Valid Inequalities -- Fractional Cutting-plane Algorithm (fcpa) for Lp(f) -- The Separation Problem for F -- A Strong Cutting-plane/branch-and-bound Algorithm for Ip. -- 3. Primal and Dual Heuristic Algorithms -- A Greedy (heuristic) Algorithm for Maximizing a Set Function -- A K-interchange Heuristic for Max{c(x): X Є S e Bn}. -- Dual Descent [a Greedy Algorithm for (3.2)] -- Analysis of Heuristics -- Simulated Annealing -- Simulated Annealing Algorithm for (3.5) -- Probabilistic Analysis -- 4. Decomposition Algorithms -- Solving the Lagrangian Dual by Subgradient Optimization -- Solving the Lagrangian Dual by Constraint Generation -- Benders' Decomposition -- Constraint Generation Algorithm for Mip' -- 5. Dynamic Programming -- A Dynamic Programming Algorithm for the 0-1 Knapsack Problem -- A Dynamic Programming Algorithm for the Uncapacitated Lot-size Problem (uls) -- 6. Notes -- Section II.5.1 -- Section II.5.2 -- Section II.5.3 -- Section II.5.4 -- Section II.5.5 -- 7. Exercises -- II.6: Applications of Special-purpose Algorithms.

1. Knapsack and Group Problems -- The Integer Knapsack Problem -- Dynamic Programming -- A Superadditive Dual Algorithm -- Heuristic Algorithms -- The Scaling/rounding (sr) Heuristic -- The Group Problem -- A Shortest-path Enumeration Algorithm for Ip -- The Increasing Group Algorithm -- The 0-1 Knapsack Problem -- An Algorithm for the Linear Programming Relaxation -- Primal Heuristic Algorithm -- Branch-and-bound -- 2. 0-1 Integer Programming Problems -- A Simplex-based Heuristic for Bip -- Algorithm -- An Fcp/branch-and-bound Algorithm -- Set Covering and Packing -- Greedy Heuristic for Set Covering -- 3. The Symmetric Traveling Salesman Problem -- Relaxations -- Primal Heuristics -- Relaxation/branch-and-bound Algorithms -- Shrinking an Edge E = (i,j) of G(x*) with Xe*= 1 -- Solution of a Large Problem -- 4. Fixed-charge Network Flow Problems -- A Branch-and-bound Algorithm for Fn -- An Fcpa for Fn -- Separation Algorithm for Generalized Flow Cover Inequalities -- Solving a Fixed-charge Uncapacitated Transportation Problem by an Fcpa and Branch-and-bound -- A Reformulation of the Single Source Problem (sfn) -- Steiner Branchings -- Reformulations Ofthe Uncapacitated Lot-size Problem (uls) -- 5. Applications of Basis Reduction -- The Subset Sum Problem -- The Reduced Basis Algorithm to Find a Solution of (5.1) -- The Linear Inequality Integer Feasibility Problem -- 6. Notes -- Section II.6.1 -- Section II.6.2 -- Section II.6.3 -- Section II.6.4 -- Section II.6.5 -- 7. Exercises -- Part III: Combinatorial Optimization -- III.1: Integral Polyhedra -- 1. Introduction -- 2. Totally Unimodular Matrices -- 3. Network Matrices -- Algorithm for Recognizing an Ept Matrix -- 4. Balanced and Totally Balanced Matrices -- Algorithm for Dfc and Fc for Row Inclusion Matrices -- Algorithm for Recognizing Tb Matrices -- 5. Node Packing and Perfect Graphs.

Algorithm for Dfnp and Fnp for Chordal Graphs.

Rave reviews for INTEGER AND COMBINATORIAL OPTIMIZATION "This book provides an excellent introduction and survey of traditional fields of combinatorial optimization . . . It is indeed one of the best and most complete texts on combinatorial optimization . . . available. [And] with more than 700 entries, [it] has quite an exhaustive reference list."-Optima "A unifying approach to optimization problems is to formulate them like linear programming problems, while restricting some or all of the variables to the integers. This book is an encyclopedic resource for such formulations, as well as for understanding the structure of and solving the resulting integer programming problems."-Computing Reviews "[This book] can serve as a basis for various graduate courses on discrete optimization as well as a reference book for researchers and practitioners."-Mathematical Reviews "This comprehensive and wide-ranging book will undoubtedly become a standard reference book for all those in the field of combinatorial optimization."-Bulletin of the London Mathematical Society "This text should be required reading for anybody who intends to do research in this area or even just to keep abreast of developments."-Times Higher Education Supplement, London Also of interest . . . INTEGER PROGRAMMING Laurence A. Wolsey Comprehensive and self-contained, this intermediate-level guide to integer programming provides readers with clear, up-to-date explanations on why some problems are difficult to solve, how techniques can be reformulated to give better results, and how mixed integer programming systems can be used more effectively. 1998 (0-471-28366-5) 260 pp.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2019. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.

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